Tagged Questions

0
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0answers
52 views

Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns

Exercise from Saff & Snider's Complex Analysis: How to determine the analyticity of this function, without using the Cauchy-Riemann equations? I tried to work from first principles (taking the ...
2
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0answers
35 views

Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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votes
1answer
68 views

Prove the inequality?

Let $f$ be an analytic function in the unit disc without zeros satisfying $|f|\leqq 1$. Prove that $$ \sup_{|z\leqq{1/5}|}|f(z)|^2\leqq \inf_{|z|\leqq{1/7}}|f(z)| $$ Help me please. These questions ...
2
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1answer
152 views

Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.

I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
3
votes
3answers
198 views

Analytic function f constant if $f(z) = 0$ or $f'(z) = 0$ for all $z$.

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be analytic and suppose that for all $z \in \mathbb{C}$, at least one of $f(z)$ and $f'(z)$ is equal to 0. Proof that $f$ is constant. Any ideas? Thanks.
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0answers
89 views

Phragmen-Lindelof theorem, question from Conway, chapter VI

Page 141, Question 3: Let $G=\{z:|\operatorname{Im} z| < \pi/2\}$ and suppose $f:G\rightarrow C$ and $\limsup|f(z)| \leq M$ on $w$ in the boundary of $G$. Also, suppose $A < \infty$ and $a ...